<![CDATA[Understanding ratio as a multiplicative relationship remains a persistent challenge in elementary mathematics, particularly in making fair judgments about situations. This study examines how sixth-grade students’ self-constructed visualizations reflected differences in their understanding of ratios. Using an exploratory qualitative design, the study involved 36 sixth-grade students from an Indonesian elementary school who had previously encountered ratio tables, double number lines, and bar models through an RME-oriented teaching experiment. Students worked individually on a fair-sharing task involving dividing a fitness trainer’s fee based on shared and individual participation time. Written responses were analyzed to identify patterns of visual representation and ratio understanding, and six focal students were purposively selected for semi-structured interviews. The analysis focused on students’ understanding of unit ratio, equivalent ratio, part-to-part, and part-to-whole. The findings showed three fairness solution patterns: proportional, two-part, and separate-payment solutions. Ratio tables supported unit- and equivalent-ratio reasoning but did not always make the shared-time structure explicit; the double-number line supported equivalent time-cost matching but allowed double-counting of shared time; and the bar model made shared and individual time more visible, supporting clearer part-to-whole coordination. The study shows how students’ self-constructed visualizations can reveal different levels of ratio understanding in fair-sharing contexts. Understanding ratio as a multiplicative relationship remains a persistent challenge in elementary mathematics, particularly in making fair judgments about situations. This study examines how sixth-grade students’ self-constructed visualizations reflected differences in their understanding of ratios. Using an exploratory qualitative design, the study involved 36 sixth-grade students from an Indonesian elementary school who had previously encountered ratio tables, double number lines, and bar models through an RME-oriented teaching experiment. Students worked individually on a fair-sharing task involving dividing a fitness trainer’s fee based on shared and individual participation time. Written responses were analyzed to identify patterns of visual representation and ratio understanding, and six focal students were purposively selected for semi-structured interviews. The analysis focused on students’ understanding of unit ratio, equivalent ratio, part-to-part, and part-to-whole. The findings showed three fairness solution patterns: proportional, two-part, and separate-payment solutions. Ratio tables supported unit- and equivalent-ratio reasoning but did not always make the shared-time structure explicit; the double-number line supported equivalent time-cost matching but allowed double-counting of shared time; and the bar model made shared and individual time more visible, supporting clearer part-to-whole coordination. The study shows how students’ self-constructed visualizations can reveal different levels of ratio understanding in fair-sharing contexts.]]>
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